5,575 research outputs found
Spin effects on the semiclassical trajectories of Dirac electrons
The relativistic semiclassical evolution of the position of an electron in
the presence of an external electromagnetic field is studied in terms of a
Newton equation that incorporates spin effects directly. This equation emerges
from the Dirac equation and allows the identification of scenarios where spin
effects are necessary to understand the main characteristics of the electron
trajectories. It involves the eigenvalues of the non-Hermitian operator
with and as the spin
and electromagnetic tensors. The formalism allows a deeper understanding on the
physics behind known analytical solutions of the Dirac equation when
translational dynamics decouples from spin evolution. As an illustrative
example, it is applied to an electron immersed in an electromagnetic field
which exhibits chiral symmetry and optical vortices. It is shown that the
polarization of intense structured light beams can be used to suppress or
enhance spin effects on the electron semiclassical trajectory; the latter case
yields a realization of a Stern-Gerlach apparatus for an electronComment: 18 pages, 3 figure
Control of atomic transition rates via laser light shaping
A modular systematic analysis of the feasibility of modifying atomic
transition rates by tailoring the electromagnetic field of an external coherent
light source is presented. The formalism considers both the center of mass and
internal degrees of freedom of the atom, and all properties of the field:
frequency, angular spectrum, and polarization. General features of recoil
effects for internal forbidden transitions are discussed. A comparative
analysis of different structured light sources is explicitly worked out. It
includes spherical waves, Gaussian beams, Laguerre-Gaussian beams, and
propagation invariant beams with closed analytical expressions. It is shown
that increments in the order of magnitude of the transition rates for Gaussian
and Laguerre-Gaussian beams, with respect to those obtained in the paraxial
limit, requires waists of the order of the wavelength, while propagation
invariant modes may considerably enhance transition rates under more favorable
conditions. For transitions that can be naturally described as modifications of
the atomic angular momentum, this enhancement is maximal (within propagation
invariant beams) for Bessel modes, Mathieu modes can be used to entangle the
internal and center of mass involved states, and Weber beams suppress this kind
of transitions unless they have a significant component of odd modes. However,
if a recoil effect of the transition with an adequate symmetry is allowed, the
global transition rate (center of mass and internal motion) can also be
enhanced using Weber modes. The global analysis presented reinforces the idea
that a better control of the transitions between internal atomic states
requires both a proper control of the available states of the atomic center of
mass, and shaping of the background electromagnetic field.Comment: 23 pages, 3 figure
Implications of maximal Jarlskog invariant and maximal CP violation
We argue here why CP violating phase Phi in the quark mixing matrix is
maximal, that is, Phi=90 degrees. In the Standard Model CP violation is related
to the Jarlskog invariant J, which can be obtained from non commuting Hermitian
mass matrices. In this article we derive the conditions to have Hermitian mass
matrices which give maximal Jarlskog invariant J and maximal CP violating phase
Phi. We find that all squared moduli of the quark mixing elements have a
singular point when the CP violation phase Phi takes the value Phi=90 degrees.
This special feature of the Jarlskog invariant J and the quark mixing matrix is
a clear and precise indication that CP violating Phase Phi is maximal in order
to let nature treat democratically all of the quark mixing matrix moduli.Comment: 25 pages, 3 figures, revte
Invariants of the harmonic conformal class of an asymptotically flat manifold
Consider an asymptotically flat Riemannian manifold of dimension with nonempty compact boundary. We recall the harmonic conformal class
of the metric, which consists of all conformal rescalings given by a
harmonic function raised to an appropriate power. The geometric significance is
that every metric in has the same pointwise sign of scalar curvature.
For this reason, the harmonic conformal class appears in the study of general
relativity, where scalar curvature is related to energy density. Our purpose is
to introduce and study invariants of the harmonic conformal class. These
invariants are closely related to constrained geometric optimization problems
involving hypersurface area-minimizers and the ADM mass. In the final section,
we discuss possible applications of the invariants and their relationship with
zero area singularities and the positive mass theorem.Comment: 26 pages, 2 figure
On the lower semicontinuity of the ADM mass
The ADM mass, viewed as a functional on the space of asymptotically flat
Riemannian metrics of nonnegative scalar curvature, fails to be continuous for
many natural topologies. In this paper we prove that lower semicontinuity holds
in natural settings: first, for pointed Cheeger--Gromov convergence (without
any symmetry assumptions) for , and second, assuming rotational symmetry,
for weak convergence of the associated canonical embeddings into Euclidean
space, for . We also apply recent results of LeFloch and Sormani to
deal with the rotationally symmetric case, with respect to a pointed type of
intrinsic flat convergence. We provide several examples, one of which
demonstrates that the positive mass theorem is implied by a statement of the
lower semicontinuity of the ADM mass.Comment: 18 pages, 4 figure
Smoothing the Bartnik boundary conditions and other results on Bartnik's quasi-local mass
Quite a number of distinct versions of Bartnik's definition of quasi-local
mass appear in the literature, and it is not a priori clear that any of them
produce the same value in general. In this paper we make progress on
reconciling these definitions. The source of discrepancies is two-fold: the
choice of boundary conditions (of which there are three variants) and the
non-degeneracy or "no-horizon" condition (at least six variants). To address
the boundary conditions, we show that given a 3-dimensional region of
nonnegative scalar curvature () extended in a Lipschitz fashion
across to an asymptotically flat 3-manifold with
(also holding distributionally along ), there exists a
smoothing, arbitrarily small in norm, such that and the
geometry of are preserved, and the ADM mass changes only by a small
amount. With this we are able to show that the three boundary conditions yield
equivalent Bartnik masses for two reasonable non-degeneracy conditions. We also
discuss subtleties pertaining to the various non-degeneracy conditions and
produce a nontrivial inequality between a no-horizon version of the Bartnik
mass and Bray's replacement of this with the outward-minimizing condition.Comment: 25 pages, 3 figure
Lower semicontinuity of the ADM mass in dimensions two through seven
The semicontinuity phenomenon of the ADM mass under pointed (i.e., local)
convergence of asymptotically flat metrics is of interest because of its
connections to nonnegative scalar curvature, the positive mass theorem, and
Bartnik's mass-minimization problem in general relativity. In this paper, we
extend a previously known semicontinuity result in dimension three for
pointed convergence to higher dimensions, up through seven, using recent work
of S. McCormick and P. Miao (which itself builds on the Riemannian Penrose
inequality of H. Bray and D. Lee). For a technical reason, we restrict to the
case in which the limit space is asymptotically Schwarzschild. In a separate
result, we show that semicontinuity holds under weighted, rather than pointed,
convergence, in all dimensions , with a simpler proof
independent of the positive mass theorem. Finally, we also address the
two-dimensional case for pointed convergence, in which the asymptotic cone
angle assumes the role of the ADM mass.Comment: 23 pages. Comments welcome
Convergence of the probability of large deviations in a model of correlated random variables having compact-support -Gaussians as limiting distributions
We consider correlated random variables taking values in
such that, for any permutation of , the random
vectors and have the same
distribution. This distribution, which was introduced by Rodr\'iguez et al
(2008) and then generalized by Hanel et al (2009), is scale-invariant and
depends on a real parameter ( implies independence).
Putting , the distribution of approaches a
-Gaussian distribution with compact support () as
increases, after appropriate scaling. In the present article, we show that the
distribution of converges, as , to a beta distribution with
both parameters equal to . In particular, the law of large numbers does
not hold since, if , then , which is the
probability of the event (large deviation), does not converges
to zero as . For and every real , we show that
decays to zero like a power law of the form with
a subdominant term of the form . If is an
integer, we show that we can analytically find upper and lower bounds for the
difference between and its () limit. We
also show that these bounds vanish like a power law of the form with a
subdominant term of the form .Comment: 8 pages, 6 figures. Accepted for publication in Journal of
Mathematical Physic
Paradoxical probabilistic behavior for strongly correlated many-body classical systems
Using a simple probabilistic model, we illustrate that a small part of a
strongly correlated many-body classical system can show a paradoxical behavior,
namely asymptotic stochastic independence. We consider a triangular array such
that each row is a list of strongly correlated random variables. The
correlations are preserved even when , since the standard central
limit theorem does not hold for this array. We show that, if we choose a fixed
number of random variables of the th row and trace over the other
variables, and then consider , the chosen ones can,
paradoxically, turn out to be independent. However, the scenario can be
different if increases with . Finally, we suggest a possible
experimental verification of our results near criticality of a second-order
phase transition.Comment: 5 pages, 7 figure
Electromagnetic waves in uniaxial crystals: General formalism with an application to Bessel beams
We present a mathematical formalism describing the propagation of a
completely general electromagnetic wave in a birefringent medium. Analytic
formulas for the refraction and reflection from a plane interface are obtained.
As a particular example, a Bessel beam impinging at an arbitrary angle is
analyzed in detail. Some numerical results showing the formation and
destruction of optical vortices are presented.Comment: 18 pages, 6 figure
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